The Harder Narasimhan type (in the sense of Gieseker semistability) of a pure-dimensional coherent sheaf on a projective scheme is known to vary semi-continuously in a flat family, which gives the well-known Harder Narasimhan stratification of the parameter scheme of the family, by locally closed subsets.
We show that each stratum can be endowed with a natural structure of a locally closed subscheme of the parameter scheme, which enjoys an appropriate universal property.
As an application, we deduce that pure-dimensional coherent sheaves of any given Harder Narasimhan type form an Artin algebraic stack.
As another application - jointly with L. Brambila-Paz and O. Mata - we describe moduli schemes for certain rank 2 unstable vector bundles on a smooth projective curve, fixing some numerical data.

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